---
title: "NCERT Solutions Class 8 Maths Chapter 10 Proportional Reasoning-2"
url: https://www.swavid.com/maths/class/8/chapter/proportional-reasoning-2/ncert-solutions
dateModified: 2026-10-07T15:34:42+00:00
---

# NCERT Solutions Class 8 Maths Chapter 10 Proportional Reasoning-2

This chapter's questions cover proportional reasoning, including direct and inverse proportions, ratios with multiple terms, and interpreting or constructing pie charts.

Free PDF (10 pages): https://www.swavid.com/api/seo/pdf/ncert/maths/class-8/swavid-ncert-solutions-class-8-maths-chapter-10-proportional-reasoning-2-fb816a15f5.pdf

## Figure it Out

### Question 1

*3 marks · Short answer*

A cricket coach schedules practice sessions that include different activities in a specific ratio — time for warm-up/cool-down : time for batting : time for bowling : time for fielding :: $3 : 4 : 3 : 5$. If each session is $150$ minutes long, how much time is spent on each activity?

**Solution**

1. Sum of the ratio terms = $3 + 4 + 3 + 5 = 15$.
2. Time for warm-up/cool-down = $150 \times \frac{3}{15} = 30$ minutes.
3. Time for batting = $150 \times \frac{4}{15} = 40$ minutes, time for bowling = $150 \times \frac{3}{15} = 30$ minutes, and time for fielding = $150 \times \frac{5}{15} = 50$ minutes.

**Answer:** Warm-up/cool-down: 30 minutes, Batting: 40 minutes, Bowling: 30 minutes, Fielding: 50 minutes.

> Common mistake: Adding the ratio terms incorrectly.

### Question 2

*3 marks · Short answer*

A school library has books in different languages in the following ratio — no. of Odiya books : no. of Hindi books : no. of English books :: $3 : 2 : 1$. If the library has $288$ Odiya books, how many Hindi and English books does it have?

**Solution**

1. The ratio of Odiya, Hindi, and English books is $3 : 2 : 1$, and the number of Odiya books corresponds to $3$ parts.
2. Since $3$ parts equal $288$ books, $1$ part = $288 \div 3 = 96$ books.
3. Number of Hindi books = $2 \times 96 = 192$ and number of English books = $1 \times 96 = 96$.

**Answer:** Hindi books: 192, English books: 96.

> Common mistake: Dividing the total by the sum of ratios instead of using the given single quantity value.

### Question 3

*3 marks · Short answer*

I have $100$ coins in the ratio — no. of ₹$10$ coins : no. of ₹$5$ coins : no. of ₹$2$ coins : no. of ₹$1$ coins :: $4 : 3 : 2 : 1$. How much money do I have in coins?

**Solution**

1. Sum of the terms of the ratio = $4 + 3 + 2 + 1 = 10$
2. Number of ₹10 coins = $100 \times \frac{4}{10} = 40$, number of ₹5 coins = $100 \times \frac{3}{10} = 30$, number of ₹2 coins = $100 \times \frac{2}{10} = 20$, and number of ₹1 coins = $100 \times \frac{1}{10} = 10$
3. Total money = $(40 \times 10) + (30 \times 5) + (20 \times 2) + (10 \times 1) = 400 + 150 + 40 + 10 = \text{₹}600$

**Answer:** ₹600

> Common mistake: Multiplying the ratio numbers directly instead of finding the number of coins first.

### Question 4

*3 marks · Short answer*

Construct a triangle with sidelengths in the ratio $3 : 4 : 5$. Will all the triangles drawn with this ratio of sidelengths be congruent to each other? Why or why not?

**Solution**

1. Yes, we can construct a triangle with sidelengths in the ratio $3 : 4 : 5$ (for example, sides $3$ cm, $4$ cm, and $5$ cm).
2. No, all triangles drawn with this ratio of sidelengths will not be congruent to each other.
3. They will be similar triangles of different sizes, as the ratio only defines the proportion of the sides, not their exact lengths.

**Answer:** No, they will not be congruent because the ratio defines similarity, not a fixed size.

> Common mistake: Confusing similar figures with congruent figures.

### Question 5

*3 marks · Short answer*

Can you construct a triangle with sidelengths in the ratio $1 : 3 : 5$? Why or why not?

**Solution**

1. No, we cannot construct a triangle with sidelengths in the ratio $1 : 3 : 5$.
2. Let the sides be $1$ unit, $3$ units, and $5$ units.
3. According to the triangle inequality property, the sum of any two sides of a triangle must be greater than the third side, but here $1 + 3 = 4$, which is less than $5$.

**Answer:** No, because the sum of the two smaller sides ($1 + 3 = 4$) is less than the third side ($5$).

> Common mistake: Attempting to construct the triangle without checking the triangle inequality condition.

## Figure it Out

### Question 1

*3 marks · Short answer*

A group of $360$ people were asked to vote for their favourite season from the three seasons — rainy, winter and summer. $90$ liked the summer season, $120$ liked the rainy season, and the rest liked the winter. Draw a pie chart to show this information.

**Solution**

1. Summer = 90 people, Rainy = 120 people, and Winter = 360 - (90 + 120) = 150 people.
2. Angle for Summer = $\frac{90}{360} \times 360^\circ = 90^\circ$, Angle for Rainy = $\frac{120}{360} \times 360^\circ = 120^\circ$, Angle for Winter = $\frac{150}{360} \times 360^\circ = 150^\circ$.
3. Diagram: Draw a circle, mark the centre and a radius, then draw sectors with angles 90°, 120°, and 150° successively, and label them with the respective seasons.

**Answer:** Sectors with angles 90°, 120°, and 150° representing summer, rainy, and winter seasons respectively.

> Common mistake: Calculating incorrect central angles by dividing by the wrong total.

### Question 2

*3 marks · Short answer*

Draw a pie chart based on the following information about viewers' favourite type of TV channel: Entertainment — $50\%$, Sports — $25\%$, News — $15\%$, Information — $10\%$.

**Solution**

1. Entertainment = 50% = $\frac{50}{100} \times 360^\circ = 180^\circ$.
2. Sports = 25% = $\frac{25}{100} \times 360^\circ = 90^\circ$, News = 15% = $\frac{15}{100} \times 360^\circ = 54^\circ$, Information = 10% = $\frac{10}{100} \times 360^\circ = 36^\circ$.
3. Diagram: Draw a circle with a radius and construct sectors of angles 180°, 90°, 54°, and 36°, then shade and label them with the channel types.

**Answer:** Sectors with angles 180°, 90°, 54°, and 36° representing Entertainment, Sports, News, and Information channels.

> Common mistake: Multiplying percentages by 100 instead of 360° to find angles.

### Question 3

*Activity*

Prepare a pie chart that shows the favourite subjects of the students in your class. You can collect the data of the number of students for each subject shown in the table (each student should choose only one subject). Then write these numbers in the table and construct a pie chart: Subject: Language, Arts Education, Vocational Education, Social Science, Physical Education, Maths, Science

**Solution**

1. Collect the number of students choosing each subject in the class to form a frequency table.
2. Calculate the central angle for each subject by dividing the number of students for that subject by the total number of students and multiplying by 360°.
3. Draw a circle and construct sectors corresponding to each calculated angle to complete the pie chart.

**Answer:** A completed frequency table and a pie chart showing the favourite subjects of students in the class.

## Figure it Out

### Question 1

*3 marks · Short answer*

Which of these are in inverse proportion?
(i) x: 40, 80, 25, 16 | y: 20, 10, 32, 50
(ii) x: 40, 80, 25, 16 | y: 20, 10, 12.5, 8
(iii) x: 30, 90, 150, 10 | y: 15, 5, 3, 45

**Part (i)**

1. Find the products of corresponding values of $x$ and $y$: $40 \times 20 = 800$, $80 \times 10 = 800$, $25 \times 32 = 800$, and $16 \times 50 = 800$.
2. Since the product $xy$ is constant ($k = 800$) for all pairs, the quantities are in inverse proportion.

Answer (i): Yes, it is in inverse proportion.

**Part (ii)**

1. Find the products of corresponding values of $x$ and $y$: $40 \times 20 = 800$, $80 \times 10 = 800$, $25 \times 12.5 = 312.5$, and $16 \times 8 = 128$.
2. Since the product $xy$ is not constant, the quantities are not in inverse proportion.

Answer (ii): No, it is not in inverse proportion.

**Part (iii)**

1. Find the products of corresponding values of $x$ and $y$: $30 \times 15 = 450$, $90 \times 5 = 450$, $150 \times 3 = 450$, and $10 \times 45 = 450$.
2. Since the product $xy$ is constant ($k = 450$) for all pairs, the quantities are in inverse proportion.

Answer (iii): Yes, it is in inverse proportion.

**Answer:** (i) and (iii) are in inverse proportion.

> Common mistake: Checking ratios instead of products to test for inverse proportion.

### Question 2

*3 marks · Short answer*

Fill in the empty cells if x and y are in inverse proportion.
x: 16, 12, _, 36
y: 9, _, 48, _

**Solution**

1. Find the constant product $k = x_1 y_1 = 16 \times 9 = 144$.
2. For $x = 12$, find $y$: $12 \times y = 144 \implies y = \frac{144}{12} = 12$.
3. For $y = 48$, find $x$: $x \times 48 = 144 \implies x = \frac{144}{48} = 3$.
4. For $x = 36$, find $y$: $36 \times y = 144 \implies y = \frac{144}{36} = 4$.

**Answer:** The missing values are 12 for the first y, 3 for the second x, and 4 for the second y.

> Common mistake: Using direct proportion relation $\frac{x_1}{y_1} = \frac{x_2}{y_2}$ instead of the inverse proportion relation $x_1 y_1 = x_2 y_2$.

## Figure it Out

### Question 1

*3 marks · Short answer*

Which of the following pairs of quantities are in inverse proportion?
(i) The number of taps filling a water tank and the time taken to fill it.
(ii) The number of painters hired and the days needed to paint a wall of fixed size.
(iii) The distance a car can travel and the amount of petrol in the tank.
(iv) The speed of a cyclist and the time taken to cover a fixed route.
(v) The length of cloth bought and the price paid at a fixed rate per metre.
(vi) The number of pages in a book and the time required to read it at a fixed reading speed.

**Solution**

1. (i) Inverse proportion, because more taps mean less time to fill the tank.
2. (ii) Inverse proportion, because more painters mean fewer days to paint the wall.
3. (iii) Direct proportion, because more petrol allows the car to travel more distance.
4. (iv) Inverse proportion, because higher speed means less time to cover a fixed route.
5. (v) Direct proportion, because buying more cloth costs more money.
6. (vi) Direct proportion, because more pages take more time to read at a fixed speed.

**Answer:** (i), (ii), and (iv) are in inverse proportion; (iii), (v), and (vi) are in direct proportion.

> Common mistake: Confusing direct proportion with inverse proportion when one quantity increases.

### Question 2

*3 marks · Short answer*

If $24$ pencils cost ₹$120$, how much will $20$ such pencils cost?

**Solution**

1. Given: Cost of $24$ pencils = ₹$120$.
2. Formula: $\frac{x_1}{y_1} = \frac{x_2}{y_2}$ for direct proportion.
3. Substitution: $\frac{24}{120} = \frac{20}{x}$
4. Result: $x = \frac{20 \times 120}{24} = \text{₹}100$.

**Answer:** ₹100

> Common mistake: Setting up the ratio inversely.

### Question 3

*3 marks · Short answer*

A tank on a building has enough water to supply $20$ families living there for $6$ days. If $10$ more families move in there, how long will the water last? What assumptions do you need to make to work out this problem?

**Solution**

1. The number of families and the number of days the water lasts are in inverse proportion.
2. Let $x$ be the number of days the water lasts for $30$ families ($20 + 10$).
3. $20 \times 6 = 30 \times x$
4. Solving for $x$, we get $x = \frac{20 \times 6}{30} = 4$ days.
5. Assumption: All families consume water at the same average rate.

**Answer:** 4 days

> Common mistake: Using direct proportion instead of inverse proportion.

### Question 4

*1 mark · Fill in the blank*

Fill in the average number of hours each living being sleeps in a day by looking at the charts. Select the appropriate hours from this list : $15, 2.5, 20, 8, 3.5, 13, 10.5, 18$.

**Solution**

1. Identify the hours corresponding to each animal from the given list based on the pie chart proportions.

**Answer:** Depends on the specific animal charts shown in the textbook figure.

> Common mistake: Misreading the fraction of the circle represented by the sector.

### Question 5

*3 marks · Short answer*

The pie chart on the right shows the result of a survey carried out to find the modes of transport used by children to go to school. Study the pie chart and answer the following questions.
(i) What is the most common mode of transport?
(ii) What fraction of children travel by car?
(iii) If $18$ children travel by car, how many children took part in the survey? How many children use taxis to travel to school?
(iv) By which two modes of transport are equal numbers of children travelling?

**Part (i)**

1. The sector with the largest angle represents the most common mode of transport.
2. Bus has the largest angle of $120^\circ$.

Answer (i): Bus

**Part (ii)**

1. Fraction of children travelling by car = $\frac{\text{Angle for car}}{\text{Total angle}}$
2. Fraction = $\frac{60^\circ}{360^\circ} = \frac{1}{6}$.

Answer (ii): $\frac{1}{6}$

**Part (iii)**

1. Let total children be $T$. $\frac{1}{6} \times T = 18$, so $T = 108$.
2. The angle for taxis is not given in the pie chart or it represents zero.

Answer (iii): 108 children took part in the survey; 0 children use taxis.

**Part (iv)**

1. Look for two sectors with equal angles in the pie chart.
2. Cycle, Car, and Two-wheeler all have $60^\circ$ each.

Answer (iv): Cycle and Car (or Two-wheeler)

**Answer:** Study the pie chart angles to answer each part.

> Common mistake: Incorrectly simplifying fractions of angles.

### Question 6

*3 marks · Short answer*

Three workers can paint a fence in $4$ days. If one more worker joins the team, how many days will it take them to finish the work? What are the assumptions you need to make?

**Solution**

1. The number of workers and the days required to complete the work are in inverse proportion.
2. Let $x$ be the number of days needed by $4$ workers ($3 + 1$).
3. $3 \times 4 = 4 \times x$
4. Solving for $x$, we get $x = 3$ days.
5. Assumption: All workers work at the same uniform rate.

**Answer:** 3 days

> Common mistake: Treating worker and day relationship as directly proportional.

### Question 7

*3 marks · Short answer*

It takes $6$ hours to fill $2$ tanks of the same size with a pump. How long will it take to fill $5$ such tanks with the same pump?

**Solution**

1. Given that $2$ tanks take $6$ hours to fill.
2. More tanks take more time, so this is a direct proportion.
3. Time taken to fill $1$ tank is $\frac{6}{2} = 3$ hours.
4. Time taken to fill $5$ such tanks is $5 \times 3 = 15$ hours.

**Answer:** $15$ hours

> Common mistake: Treating the problem as inverse proportion.

### Question 8

*3 marks · Short answer*

A given set of chairs are arranged in $25$ rows, with $12$ chairs in each row. If the chairs are rearranged with $20$ chairs in each row, how many rows does this new arrangement have?

**Solution**

1. Given that chairs are arranged in $25$ rows with $12$ chairs in each row, making the total number of chairs $25 \times 12 = 300$.
2. When rearranged with $20$ chairs in each row, let the number of rows be $x$.
3. Since the total number of chairs remains constant, more chairs per row means fewer rows, which is an inverse proportion.
4. Thus, $20 \times x = 300$, giving $x = \frac{300}{20} = 15$.

**Answer:** $15$ rows

> Common mistake: Dividing the total number of chairs by something else or using direct proportion.

### Question 9

*3 marks · Short answer*

A school has $8$ periods a day, each of $45$ minutes duration. How long is each period, if the school has $9$ periods a day, assuming that the number of school hours per day stays the same?

**Solution**

1. Total school hours in a day = $8 \times 45$ minutes = $360$ minutes.
2. If the school has $9$ periods a day and the total school hours remain the same, let the duration of each period be $x$ minutes.
3. More periods per day mean each period is shorter, which is an inverse proportion.
4. Thus, $9 \times x = 360$, giving $x = \frac{360}{9} = 40$ minutes.

**Answer:** $40$ minutes

> Common mistake: Using direct proportion instead of inverse proportion.

### Question 10

*3 marks · Short answer*

A small pump can fill a tank in $3$ hours, while a large pump can fill the same tank in $2$ hours. If both pumps are used together, how long will the tank take to fill?

**Solution**

1. The small pump fills the tank in $3$ hours, so in $1$ hour it fills $\frac{1}{3}$ of the tank.
2. The large pump fills the same tank in $2$ hours, so in $1$ hour it fills $\frac{1}{2}$ of the tank.
3. When both pumps are used together, the fraction of the tank filled in $1$ hour is $\frac{1}{3} + \frac{1}{2} = \frac{2 + 3}{6} = \frac{5}{6}$.
4. Therefore, the time taken to fill the entire tank is $\frac{6}{5}$ hours or $1$ hour and $12$ minutes.

**Answer:** $\frac{6}{5}$ hours ($1$ hour $12$ minutes)

> Common mistake: Adding the times directly as $3 + 2 = 5$ instead of adding their work rates.

### Question 11

*3 marks · Short answer*

A factory requires $42$ machines to produce a given number of toys in $63$ days. How many machines are required to produce the same number of toys in $54$ days?

**Solution**

1. Fewer days to produce the same number of toys require more machines, so the number of machines and days are inversely proportional.
2. Let $x$ be the number of machines required for $54$ days.
3. Using the inverse proportion relation, $42 \times 63 = x \times 54$.
4. Solving for $x$, we get $x = \frac{42 \times 63}{54} = \frac{2646}{54} = 49$ machines.

**Answer:** $49$ machines

> Common mistake: Treating the quantities as directly proportional.

### Question 12

*3 marks · Short answer*

A car takes $2$ hours to reach a destination, travelling at a speed of $60$ km/h. How long will the car take if it travels at a speed of $80$ km/h?

**Solution**

1. The total distance is constant. When speed increases, the time taken decreases, which is an inverse proportion.
2. Let $x$ be the time taken at a speed of $80$ km/h.
3. Using the relation $x_1 y_1 = x_2 y_2$, we have $60 \times 2 = 80 \times x$.
4. Solving for $x$, we get $x = \frac{120}{80} = \frac{3}{2} = 1.5$ hours.

**Answer:** $1.5$ hours

> Common mistake: Using direct proportion by dividing speed instead of equating distances.

## Frequently asked questions

### How many questions are there in NCERT Solutions for Class 8 Maths Chapter 10 Proportional Reasoning-2?

This chapter contains multiple 'Figure it Out' sections featuring a total of 22 questions across different exercises. You can access SwaVid's free PDF and step-by-step solutions for all these questions on this page only.

### What topics are covered in Class 8 Maths Chapter 10 Proportional Reasoning-2?

The chapter covers concepts such as dividing quantities and wholes in given ratios with multiple terms, triangle construction and similarity, and the triangle inequality property. It also includes collecting data, pie chart construction from percentages, and solving problems on direct and inverse proportions, unitary method, and work rate.

### Which are the hardest question types in this chapter and how should I approach them?

Questions involving pie chart construction from percentages and complex inverse proportion or work rate problems are generally considered the trickiest. To approach them, carefully note the given values, apply the appropriate ratio or proportion formula step-by-step, and verify your calculations.

### How can I write answers for full marks in Class 8 Maths Chapter 10?

To score full marks, always write down the given information clearly, state the formula or ratio being used, and show every intermediate calculation step. Referring to SwaVid's free PDF and step-by-step solutions available on this page only will help you understand the ideal presentation format.

### Is the free PDF for this chapter available according to the new NCERT book?

Yes, the solutions are fully updated for the new NCERT book based on the NCF 2023 for the 2026-27 session. You can find SwaVid's free PDF and step-by-step solutions right here on this page only.

## Related pages

- [Class 8 Maths chapters](https://www.swavid.com/maths/class/8)

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